About Turan`s problem(inequality) in multivariable Hi. I have a question related to Turan`s problem, that is  
Find a sequence of polynomial $P_n(x)$ satisfying $P_{n+1}(x)P_{n-1}(x) < P_{n}^2(x)$.
I am considering the generalized question for positive multivarible, i.e.
Let $k$ be a positive integer and let $x_1, ... x_k$ be k indeterminates. 
For $x_1, ... x_n > 0$, find a sequence of polynomial $P_n(x_1, ... , x_k)$ satisfying
$P_{n+1}(x_1, ... , x_k) P_{n-1}(x_1, ... , x_k)  < A(n)P_n(x_1, ... , x_k)^2$ where
$A(n)$ is some fixed function for $n$. 
Is there any result or some reference related to this problem? 
In particular, I am interested in the case when $A(n) = \frac{n+2}{n+1}$. And I tried to check the above inequality by using maple by letting $P_n(x_1, ... , x_k) = x_1^n + ... + x_k^n$. then suprisingly(for me) I didn't find a counterexample until now, nor prove the inequality. 
How can I prove (or find a counterexample) of this inequality? 
I really appriciate for your any comment and help. 
Thank you in advance.
 A: I am sorry. Ottem is right. $P_n$ is a polynomial of degree $n$. In particular, you may assume that $P_n$ is a symmetric function. Let's focus that $P_n(x_1, ... ,x_k) = x_1^n + ... x_k^n$, for any $k > 0 , n > 1$ given integers. 
By rearrangement inequality, we can easily show that $ F(n) = \frac{P_n^2}{P_{n+1}P_{n-1} \leq 1.$
I want to know about the lower bound of $F(n)$ for each $n$.  
A: Let's try this in answer form.  Let $Q = Q(x_1,\ldots,x_n)$ be a multivariate polynomial over some ordered ring containing the integers (with the usual ordering on the integers).
Then define $P_n = Q - n$.  Then
$(Q - (n+1))(Q - (n-1)) = Q^2 -2nQ + n^2 - 1 = (Q-n)^2 - 1$  .
This seems to satisfy your inequality, even with $A(n)= 1$ .  Was there something else?
I would prefer to not find references nor do more work unless you can tell me more of what you know about the problem and more specifics on what you actually want.
Gerhard "Ask Me About System Design" Paseman, 2011.02.17
A: *

*For the case of a single variable an obvious condition on polynomials is that a ratio
$$
\frac{P_{n-1}(x)P_{n+1}(x)}{(P_n(x))^2}
$$ 
is monotone. Then  sharp estimates hold true with limits via $P_n(0),P_n(\infty)$ and so polynomial's coefficients.

*Consider also a reference http://arxiv.org/abs/1207.0936
with examples for sections of exponential series.
